A matrix is defined as:
$$ \begin{bmatrix} {\bf a}^T{\bf b}_{1,1}, \ldots, {\bf a}^T{\bf b}_{1,N} \\ \cdots, \cdots, \cdots, \\ {\bf a}^T{\bf b}_{N,1}, \ldots, {\bf a}^T{\bf b}_{N,N} \end{bmatrix} $$
where ${\bf a}$ and ${\bf b}_{i,j}$ are $N \times 1$ vector. This is a matrix where each element is an inner product with the same vector. What is a compact notation (that is not element-wise) of such a matrix?
As clarified in the comments, I am looking for a matrix formulation where the vector ${\bf a}$ comes out and is not element-wise notation.